Question

A positive point charge q1 creates an electric field of magnitude e1 at a spot located at a distance r1 from the charge. the charge is replaced by another positive point charge q2, which creates a field of magnitude e2 = e1 at a distance of r2 = 2r1. how is q2 related to q1?

Answer 1

Let's answer this in a straightforward way:
The Electric field due to a point charge at a point with distance r relative to the point charge is:

`E(r)=(1)/(4\pi\epsilon_0)(q)/(r^2)`
For E1 we have is:

`E_1=(1)/(4\pi\epsilon_0)(q_1)/(r^2_1)`
For E2=E1 we have:

`E_2=(1)/(4\pi\epsilon_0)(q_2)/(r^2_2)`

We the get the following equation:

`E_1=E_2\implies (q_1)/(r^2_1)=(q_2)/(r^2_2)\implies q_2=q_1(r^2_2)/(r^2_1)`

Because
`r_2=2r_1` we get from the above expression for q2:

`q_2=q_1(r^2_2)/(r^2_1)=q_1((2r_1)^2)/(r^2_1)=q_1(4r^2_1)/(r^2_1)=4q_1`